Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $p = \dfrac{3k - 24}{10k - 30} \times \dfrac{k^2 + 3k}{k^2 - 5k - 24} $
First factor the quadratic. $p = \dfrac{3k - 24}{10k - 30} \times \dfrac{k^2 + 3k}{(k - 8)(k + 3)} $ Then factor out any other terms. $p = \dfrac{3(k - 8)}{10(k - 3)} \times \dfrac{k(k + 3)}{(k - 8)(k + 3)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ 3(k - 8) \times k(k + 3) } { 10(k - 3) \times (k - 8)(k + 3) } $ $p = \dfrac{ 3k(k - 8)(k + 3)}{ 10(k - 3)(k - 8)(k + 3)} $ Notice that $(k + 3)$ and $(k - 8)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ 3k\cancel{(k - 8)}(k + 3)}{ 10(k - 3)\cancel{(k - 8)}(k + 3)} $ We are dividing by $k - 8$ , so $k - 8 \neq 0$ Therefore, $k \neq 8$ $p = \dfrac{ 3k\cancel{(k - 8)}\cancel{(k + 3)}}{ 10(k - 3)\cancel{(k - 8)}\cancel{(k + 3)}} $ We are dividing by $k + 3$ , so $k + 3 \neq 0$ Therefore, $k \neq -3$ $p = \dfrac{3k}{10(k - 3)} ; \space k \neq 8 ; \space k \neq -3 $